Reversal of Limits MCQ Quiz - Objective Question with Answer for Reversal of Limits - Download Free PDF

\(\log A= \displaystyle \lim_{n\rightarrow \infty} \log \left (\dfrac {(n + 1) (n + 2) \ldots 3n}{n^{2n}} \right )^{\frac {1}{n}}\)

\(=\displaystyle \lim_{n\rightarrow \infty} \dfrac 1n \log \left (\dfrac {(n + 1) (n + 2) \ldots 3n}{n^{2n}} \right )\)

\(=\displaystyle \lim_{n\rightarrow \infty} \dfrac 1n \sum_{r=0}^{r=2n}\log\left (\dfrac {n+r}{n} \right )\)

Let \(a,b\) are the limits of the integration,

\(\Rightarrow\) when \(r=0\), \(a=\displaystyle \lim_{n\rightarrow \infty} \dfrac 1n =0\); and \(r=2n\), \(b=\displaystyle \lim_{n\rightarrow \infty} \dfrac {2n}{n} =2\)

\(\log A = \displaystyle \lim_{n\rightarrow \infty} \dfrac 1n \sum_{r=1}^{r=2n}\log\left(1+\dfrac rn\right)\)

\(=\displaystyle\int_{0}^{2} \log (1+x) dx\)

\(=\left[(1+x)\log(1+x) - (1+x)\right]_{0}^{2}\) [\(\because \int \log x=x(\log x-1)\)]

\(\log A = 3 \log 3 - 2\)

\(A=e^{3 \log 3 -2}\)

\(=e^{3 \log 3} . e^{-2}\)

\(=\dfrac {e^{3 \log 3}}{e^2} = \dfrac {e^{\log_e 27}}{e^2}=\dfrac {27}{e^2}\)